## Theoretical considerations in the use of scalar-tensor theories of gravity in inflationary models

##### Abstract

The inflationary paradigm is one which was designed to answer questions that
arose from classical Hot Big Bang cosmology. The period of rapid expansion in
the early Universe provides a mechanism to solve the flatness, horizon and relic
problems. More importantly, since the theory was first introduced it has been
realised that it also provides a mechanism to generate the initial perturbations
from which structure in the Universe can grow.
In the zoo of potential inflationary models there is a dominant class: slow-roll
inflation. The idea that the energy density of the inflationary field is dominated
by its potential highly simplifies the calculations required to predict observable
quantities. This simplification relies on all the information required to know the
subsequent dynamics of the field to be encoded in the space ϕ− ˙ϕ; it must be an
effective phase space. I show that ϕ−˙ϕ can be considered to be such a space for the
most general scalar-tensor theory which gives second-order equations of motion:
Horndeski theory. There are theoretical issues associated with this reduction that
are illuminated through specific examples in which they occur.
A theoretical issue with inflation is that there is an overabundance of models, with
some capable of predicting any value of the possible observables. The second
block of work in this thesis looks at a particular set of models that make the
same observational prediction. These “attractor” models utilise a non-minimal
coupling between the inflationary fields and gravity and are studied in depth,
both in the case of one and several fields.
Firstly, I examine the Universal Attractors, a single field subset of these models.
I show, in detail, the observational prediction such a model makes in the case
of a strong non-minimal coupling and then examine the constraints it would
be possible to put on such a coupling if a confirmed detection of primordial
gravitational waves was made. Despite the discussion existing in the literature
there is a small deviation of the Universal Attractor models from the predictions of
the Starobinsky model. Furthermore, the coupling, ξ is found to be constrained
so that |ξ| < 1 in the case where there a level of detectable primordial tensor
modes.
While the attractor models have an effective one-field description in reality there
are several other fields that are assumed to be fixed during the inflationary phase.
This claim requires careful examination as the field-space of the models generally
is not flat. This curvature can cause a destabilising effect with certain parameters
and so I investigate how susceptible the ξ-attractors and related models are to the
destabilisation. A key result of this chapter is to highlight how important it is to
not rely on the slow-roll approximation when assessing the effect of the instability,
as the region where the effect begins to become large corresponds with the region
where slow-roll begins to break down. Assuming the slow-roll approximation
is valid leads to an over-estimation of the effect that the instability mechanism
has. Despite this, some of the models considered are seen to experience the
instability for certain ranges of model parameters. Making the assumption that
any occurrence of the instability will, at the very least, move the observational
prediction of the model outside the currently constrained range allows a constraint
on the model parameter in question which directly translates to a theoretical lower
bound on the tensor-scalar ratio, r > 0.0005.